Definition

Norm on a Linear Space Normed Space

Functional Analysis

Definition

Inner Product

Functional Analysis

Definition

Linear Transformation/Operator

Functional Analysis A real-valued function ||x|| defined on a linear space X, where x ∈ X, is said to be a norm on X if Positivity ||x|| ≥ 0, Triangle Inequality ||x + y|| ≤ ||x|| + ||y||, Homogeneity ||αx|| = |α| ||x||, α an arbitrary scalar, Positive Definiteness ||x|| = 0 if and only if x = 0, where x and y are arbitrary points in X. A linear/vector space with a norm is called a normed space.

Let X be a complex linear space. An inner product on X is a mapping that associates to each pair of vectors x, y a scalar, denoted (x, y), that satisfies the following properties:

Additivity (x + y, z) = (x, z) + (y, z), Homogeneity (α x, y) = α(x, y), Symmetry (x, y) = \overline{(y, x)}, Positive Definiteness (x, x) > 0, when x ≠ 0.

A transformation L of (operator on) a linear space X into a linear space Y, where X and Y have the same scalar field, is said to be a linear transformation (operator) if

1. L(αx) = αL(x), ∀x ∈ X and ∀ scalars α, and

2. L(x_{1} + x_{2}) = L(x_{1}) + L(x_{2}) for all x_{1}, x_{2} ∈ X.